Theorem cvrnrefN

Description: The covers relation is not reflexive. ( cvnref analog.) (Contributed by NM, 1-Nov-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cvrne.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	cvrne.c	⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
Assertion	cvrnrefN	⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ¬ 𝑋 𝐶 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		cvrne.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		cvrne.c	⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
3		eqid	⊢ 𝑋 = 𝑋
4		simpll	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑋 ) → 𝐾 ∈ 𝐴 )
5		simplr	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑋 ) → 𝑋 ∈ 𝐵 )
6		simpr	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑋 ) → 𝑋 𝐶 𝑋 )
7	1 2	cvrne	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑋 ) → 𝑋 ≠ 𝑋 )
8	4 5 5 6 7	syl31anc	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑋 ) → 𝑋 ≠ 𝑋 )
9	8	ex	⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 𝐶 𝑋 → 𝑋 ≠ 𝑋 ) )
10	9	necon2bd	⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 = 𝑋 → ¬ 𝑋 𝐶 𝑋 ) )
11	3 10	mpi	⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ¬ 𝑋 𝐶 𝑋 )